Applied Mathematics 1 - May 2013

First Year Engineering (Semester 1)

TOTAL MARKS: 80
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any three from the remaining questions.
(3) Assume data if required.
(4) Figures to the right indicate full marks. 1(a) If cosh x = sec θ, prove that x = log (sec θ + tan θ)(3 marks) 1(b) If u = log (x²+y²), prove that
$$\dfrac{\partial^2 u}{\partial x \ \partial y} = \dfrac{\partial^2 u}{\partial y \ \partial x}$$(3 marks) 1(c) If x = r cosθ, y = r sinθ;
$$ \dfrac{\partial(x,y)}{\partial(r, \theta)} $$(3 marks) 1(d) Expand log(1 + x + x² + x³) in powers of x upto x⁸.(3 marks) 1(e) Show that every square matrix can be uniquely expressed as sum of a symmetric and Skew-symmetric matrix.(4 marks) 1(f) If y = cos x.cos 2x.cos 3x then find its n^th order derivative
(4 marks) 2(a) Solve the equation x⁶-i=0.(6 marks) 2(b) Reduce matrix A to normal form and find its rank where
$$A={ \left[ \begin{array}{ccc} 1 & 2 & 3 &2 \\ 2 & 3 & 5 & 1 \\ 1 & 3 & 4 &5 \end{array} \right]}$$(6 marks) 2(c) State and prove Euler's theorem for a homogeneous function in two variable. And hence find
$$x\dfrac{\partial u}{\partial x} + y\dfrac{\partial u}{\partial y} \ \ \ where \ \ u = \dfrac{\sqrt{x}+\sqrt{y}}{x+y}$$(8 marks) 3(a) Determine the values of λ so that the equations
x+y+z=1,
x+2y+4z= λ ,
x+4y+10z=λ²
have a solution and solve them completely in each case.(6 marks) 3(b) Find the stationary values of
x³ + y³ - 3axy, a > 0(6 marks) 3(c) Separate into real and imaginary parts
tan^-1(e^iθ)(8 marks) 4(a) If x = u cos v and y = u sin v
$$\dfrac{\partial(x,y)}{\partial(u,v)}.\dfrac{\partial(u,v)}{\partial(x,y)} =1$$(6 marks) 4(b) If tan[log(x + iy)] = a+ib,
$$prove \ that \ tan[log(x^2+y^2)]=\dfrac{2a}{1-a^2-b^2}$$
where a² + b² ≠ 1.(6 marks) 4(c) Using Gauss- Seidel iteration method solve,
10x₁ + x₂ + x₃ = 12,
2x₁ + 10x₂ + x₃=13,
2x₁ + 2x₂ + 10x₃ = 14
Upto three iterations.(8 marks) 5(a) In a series of sines of multiple of θ, expand sin⁷ θ(6 marks) 5(b) Evaluate the following:
$$\lim_{x\rightarrow 1}\dfrac{x^x-x}{x-1-logx}$$(6 marks) 5(c) Prove the following if y^1/m + y^-1/m = 2x;
(x² -1) y_(n+2) + (2n+1)xy_(n+1) + (n²-m²)y_n = 0 (8 marks) 6(a) Examine the following vectors for linear dependence/independence
X₁ = (a,b,c), X₂ = (b,c,a), X₃ = (c,a,b)
where a+b+c ≠ to zero. (6 marks) 6(b) If z = f(x,y) , x=(e^u + e^-v), y=(e^-u - e^v)
$$ \dfrac{\partial z}{\partial u}-\dfrac{\partial z}{\partial v} = x\dfrac{\partial z}{\partial x}-y\dfrac{\partial z}{\partial y}$$(6 marks) 6(c) Fit a straight line to following data and also estimate the production in 1957.